where \(\theta \in \Theta \subseteq \mathbb {R}^p\)  is the model parameter, \(\Theta \)  is an open set, \(b_\theta : \mathbb {R}^d\rightarrow \mathbb {R}^d\) , \(W=(W_t)_{t\ge 0}\)  is a \(d\) -dimensional Brownian motion and the initial value \(X_0\)  is a random vector independent of \(W\)  (here we suppress the dependence of \(X\)  on \(\theta \) ). The underlying observations are \((X_t)_{t\in [0,T]}\)  and we assume that \(p\) , \(d\)  and \(T\)  are large. Such high dimensional models find numerous applications in physics, economics and biology among other sciences. For instance, high dimensional diffusions are studied in the context of particle systems within mean field theory. Probabilistic results in this topic can be found in the classical works [1]}, [2]}, [3]} while applications of particle systems in sciences are studied in [4]}, [5]}, [6]} among many others. More recently, statistical inference for large particle systems and their mean field limits has been investigated in [7]}, [8]}, [9]}, [10]}, [11]}. However, the authors only consider a finite dimensional parameter space \(\Theta \) .
